One-dimensional central extensions \(E=\mathfrak {g}\oplus \mathbb {F}c\)  of an ordinary Lie algebra \(\mathfrak {g}\)  are parameterized by the cohomology group \(H^2(\mathfrak {g})\)  [1]}, and restricted one-dimensional central extensions of a restricted Lie algebra \(\mathfrak {g}\)  with \(c^{[p]}=0\)  are parameterized by the restricted cohomology group \(H^2_*(\mathfrak {g})\) 
[2]}. If \((\varphi ,\omega )\in C^2_*(\mathfrak {g})\)  is a restricted 2-cocycle, then the corresponding restricted one-dimensional central extension \(E=\mathfrak {g}\oplus \mathbb {F}c\)  has Lie bracket and \([p]\) -operation defined by
\([g,h] =[g,h]_{\mathfrak {g}}+\varphi (g\wedge h) c,\nonumber \\[g, c] = 0,\nonumber \\g^{[p]} = p^{[p]_{\mathfrak {g}}}+\omega (g) c,\nonumber \\c^{[p]} = 0,\) 
